| L(s) = 1 | + (−1.27 + 3.92i)2-s + (0.664 + 5.15i)3-s + (−7.33 − 5.32i)4-s + (0.616 − 0.200i)5-s + (−21.0 − 3.96i)6-s + (8.50 − 11.7i)7-s + (3.54 − 2.57i)8-s + (−26.1 + 6.84i)9-s + 2.67i·10-s + (16.7 + 32.3i)11-s + (22.5 − 41.3i)12-s + (54.2 + 17.6i)13-s + (35.1 + 48.3i)14-s + (1.44 + 3.04i)15-s + (−16.8 − 51.7i)16-s + (−5.39 − 16.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.451 + 1.38i)2-s + (0.127 + 0.991i)3-s + (−0.916 − 0.665i)4-s + (0.0551 − 0.0179i)5-s + (−1.43 − 0.270i)6-s + (0.459 − 0.632i)7-s + (0.156 − 0.113i)8-s + (−0.967 + 0.253i)9-s + 0.0846i·10-s + (0.459 + 0.887i)11-s + (0.543 − 0.993i)12-s + (1.15 + 0.376i)13-s + (0.670 + 0.923i)14-s + (0.0248 + 0.0523i)15-s + (−0.262 − 0.808i)16-s + (−0.0769 − 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.182575 + 1.00785i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.182575 + 1.00785i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.664 - 5.15i)T \) |
| 11 | \( 1 + (-16.7 - 32.3i)T \) |
| good | 2 | \( 1 + (1.27 - 3.92i)T + (-6.47 - 4.70i)T^{2} \) |
| 5 | \( 1 + (-0.616 + 0.200i)T + (101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (-8.50 + 11.7i)T + (-105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-54.2 - 17.6i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (5.39 + 16.5i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-62.1 - 85.4i)T + (-2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + 145. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-26.1 - 19.0i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-80.1 + 246. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-120. - 87.7i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-246. + 179. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 267. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (147. + 203. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (225. + 73.3i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (394. - 543. i)T + (-6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-188. + 61.1i)T + (1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + 557.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-226. + 73.6i)T + (2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (420. - 578. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (973. + 316. i)T + (3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (278. + 857. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 165. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-158. + 487. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.63297509739926914204835945372, −15.72657232420817211876392014102, −14.69978052998909267493440659968, −13.86997238062127981630762514217, −11.57627276028000046287031667971, −10.04781453960966058139761978241, −8.864637709204605279993527656374, −7.61604330282450838736385765018, −5.98537108869656324299139203625, −4.30484362553940406749869714992,
1.27365000699255690642011968822, 3.09161522860825004543802797328, 6.08947314368302168211237129683, 8.212379889545375529000010155607, 9.241254153773777665987310335008, 11.12789232851320970177148576682, 11.70160843558563452243629960628, 12.98319917232831889188461865073, 13.98218544725243891038183987097, 15.72247465200859589488175794609
